Abstract

When computing descriptors of image data,
the type of information that can be extracted
may be strongly dependent on the scales
at which the image operators are applied.
This article presents a systematic methodology for
addressing this problem.
A mechanism is presented for
automatic selection of scale levels
when detecting one-dimensional image features,
such as edges and ridges.

A concept of a scale-space edge
is introduced, defined as a connected set of points in scale-space
at which:
(i) the gradient magnitude assumes a local maximum in
the gradient direction, and
(ii) a normalized measure of the strength of the edge
response is locally maximal over scales.
An important consequence of this definition is that
it allows the scale levels to vary along the edge.

Two specific measures of edge strength are analysed in detail,
the gradient magnitude and a differential expression derived from
the third-order derivative in the gradient direction.
For a certain way of normalizing these differential descriptors,
by expressing them in terms of so-called
gamma-normalized derivatives,
an immediate consequence of this definition is that the edge detector will
adapt its scale levels to the local image structure.
Specifically, sharp edges will be detected at fine scales
so as to reduce the shape distortions due to scale-space smoothing,
whereas sufficiently coarse scales will be selected at diffuse edges,
such that an edge model is a valid abstraction of the
intensity profile across the edge.

Since the scale-space edge is defined from the intersection of
two zero-crossing surfaces in scale-space,
the edges will by definition form closed curves.
This simplifies selection of salient edges,
and a novel significance measure is proposed,
by integrating the edge strength along the edge.
Moreover, the scale information associated with each edge
provides useful clues to the physical nature of the edge.

With just slight modifications, similar ideas can be used for formulating
ridge detectors with automatic selection,
having the characteristic property that
the selected scales on a scale-space ridge
instead reflect the width of the ridge.

It is shown how the methodology can be
implemented in terms of straightforward visual front-end operations,
and the validity of the approach is supported by theoretical analysis as
well as experiments on real-world and synthetic data.